Adding and comparing kilometers and hours in a system of equations

The 5 and 60 are in km/h, so when you multiply the second equation by 5, you are actually multiplying by 5 km/h, which makes the units make sense. You should try to either write down all the units, like for x, y, 5, and 60 in this case, or none at all.
If you divide the first equation by 1 kilometer and the second equation  by 1 hour you will get a pair of dimensionless equations. This is why we can ignore the units when we manipulate the equations. I think that she will like this explanation.
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The units don't matter. If you divide the first eqn by 1 km and the second eqn by 1 hour you have two eqns without units.

Going forward, I would omit the units when writing down the eqns.
Tell her that each equation on its own must be consistent with regard to the units, but the arithmetic we use to solve the system does not require this consistency. Each eqn represents a different relationship between x and y.
Tell her that x and y have the units of kilometers in both equations
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